Low-energy four-dimensional effective field theories arising from compactified theories of gravity involve towers of interacting spin-0 and spin-2 fields (and potentially spin-1 fields as well, though these are often eliminated by imposing an orbifold symmetry on the compact extra dimension). The massive spin-2 resonances – sometimes called Kaluza-Klein (KK) gravitons – are particularly interesting. The existence of self interactions between these KK gravitons is problematic because typically scattering amplitudes between massive spin-2 particles grow far too rapidly with energy to keep unitarity constraints satisfied much beyond the mass of lightest massive spin-2 state involved. For example, theories of massive gravity that extend 4D general relativity by adding a Fierz-Pauli mass term [1]} result in 2-to-2 scattering amplitudes for the helicity-zero channel (the channel whose amplitude has the highest energy-growth) that grow like \(s^{5}/(m_{\text{FP}}^{8}M_{\text{Pl}}^{2})\)  [2]}, where \(m_{\text{FP}}\)  is the mass of the graviton, \(s\)  the squared center-of-mass energy, and \(M_{\text{Pl}}\)  the reduced Planck mass. Adding carefully-chosen potential terms to the model [3]}, [4]} can soften matrix element growth down to \({\cal O}(s^3)\) . However, compactified theories of extra-dimensional gravity should – due to higher-dimensional diffeomorphism invariance, requiring a smooth \(m_{\text{KK}}\rightarrow 0\)  limit – grow at most as \({\cal O}(s)\) .
